## Linear Feedback Shift Registers for the Uninitiated, Part VIII: Matrix Methods and State Recovery

Last time we looked at a dsPIC implementation of LFSR updates. Now we’re going to go back to basics and look at some matrix methods, which is the third approach to represent LFSRs that I mentioned in Part I. And we’re going to explore the problem of converting from LFSR output to LFSR state.

Matrices: Beloved Historical DregsElwyn Berlekamp’s 1966 paper Non-Binary BCH Encoding covers some work on

## Linear Feedback Shift Registers for the Uninitiated, Part VII: LFSR Implementations, Idiomatic C, and Compiler Explorer

The last four articles were on algorithms used to compute with finite fields and shift registers:

- multiplicative inverse
- discrete logarithm
- determining characteristic polynomial from the LFSR output

Today we’re going to come back down to earth and show how to implement LFSR updates on a microcontroller. We’ll also talk a little bit about something called “idiomatic C” and a neat online tool for experimenting with the C compiler.

## Lazy Properties in Python Using Descriptors

This is a bit of a side tangent from my normal at-least-vaguely-embedded-related articles, but I wanted to share a moment of enlightenment I had recently about descriptors in Python. The easiest way to explain a descriptor is a way to outsource attribute lookup and modification.

Python has a bunch of “magic” methods that are hooks into various object-oriented mechanisms that let you do all sorts of ridiculously clever things. Whether or not they’re a good idea is another...

## Linear Feedback Shift Registers for the Uninitiated, Part VI: Sing Along with the Berlekamp-Massey Algorithm

The last two articles were on discrete logarithms in finite fields — in practical terms, how to take the state \( S \) of an LFSR and its characteristic polynomial \( p(x) \) and figure out how many shift steps are required to go from the state 000...001 to \( S \). If we consider \( S \) as a polynomial bit vector such that \( S = x^k \bmod p(x) \), then this is equivalent to the task of figuring out \( k \) from \( S \) and \( p(x) \).

This time we’re tackling something...

## Linear Feedback Shift Registers for the Uninitiated, Part V: Difficult Discrete Logarithms and Pollard's Kangaroo Method

Last time we talked about discrete logarithms which are easy when the group in question has an order which is a smooth number, namely the product of small prime factors. Just as a reminder, the goal here is to find \( k \) if you are given some finite multiplicative group (or a finite field, since it has a multiplicative group) with elements \( y \) and \( g \), and you know you can express \( y = g^k \) for some unknown integer \( k \). The value \( k \) is the discrete logarithm of \( y \)...

## Linear Feedback Shift Registers for the Uninitiated, Part IV: Easy Discrete Logarithms and the Silver-Pohlig-Hellman Algorithm

Last time we talked about the multiplicative inverse in finite fields, which is rather boring and mundane, and has an easy solution with Blankinship’s algorithm.

Discrete logarithms, on the other hand, are much more interesting, and this article covers only the tip of the iceberg.

What is a Discrete Logarithm, Anyway?Regular logarithms are something that you’re probably familiar with: let’s say you have some number \( y = b^x \) and you know \( y \) and \( b \) but...

## Linear Feedback Shift Registers for the Uninitiated, Part III: Multiplicative Inverse, and Blankinship's Algorithm

Last time we talked about basic arithmetic operations in the finite field \( GF(2)[x]/p(x) \) — addition, multiplication, raising to a power, shift-left and shift-right — as well as how to determine whether a polynomial \( p(x) \) is primitive. If a polynomial \( p(x) \) is primitive, it can be used to define an LFSR with coefficients that correspond to the 1 terms in \( p(x) \), that has maximal length of \( 2^N-1 \), covering all bit patterns except the all-zero...

## Linear Feedback Shift Registers for the Uninitiated, Part II: libgf2 and Primitive Polynomials

Last time, we looked at the basics of LFSRs and finite fields formed by the quotient ring \( GF(2)[x]/p(x) \).

LFSRs can be described by a list of binary coefficients, sometimes referred as the polynomial, since they correspond directly to the characteristic polynomial of the quotient ring.

Today we’re going to look at how to perform certain practical calculations in these finite fields. I maintain a Python library on bitbucket called...

## Linear Feedback Shift Registers for the Uninitiated, Part I: Ex-Pralite Monks and Finite Fields

Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.

— Évariste Galois, May 29, 1832

I was going to call this short series of articles “LFSRs for Dummies”, but thought better of it. What is a linear feedback shift register? If you want the short answer, the Wikipedia article is a decent introduction. But these articles are aimed at those of you who want a little bit deeper mathematical understanding,...

## Ten Little Algorithms, Part 6: Green’s Theorem and Swept-Area Detection

Other articles in this series:

- Part 1: Russian Peasant Multiplication
- Part 2: The Single-Pole Low-Pass Filter
- Part 3: Welford's Method (And Friends)
- Part 4: Topological Sort
- Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method

This article is mainly an excuse to scribble down some cryptic-looking mathematics — Don’t panic! Close your eyes and scroll down if you feel nauseous — and...

## Lazy Properties in Python Using Descriptors

This is a bit of a side tangent from my normal at-least-vaguely-embedded-related articles, but I wanted to share a moment of enlightenment I had recently about descriptors in Python. The easiest way to explain a descriptor is a way to outsource attribute lookup and modification.

Python has a bunch of “magic” methods that are hooks into various object-oriented mechanisms that let you do all sorts of ridiculously clever things. Whether or not they’re a good idea is another...

## Adventures in Signal Processing with Python

Author’s note: This article was originally called Adventures in Signal Processing with Python (MATLAB? We don’t need no stinkin' MATLAB!) — the allusion to The Treasure of the Sierra Madre has been removed, in deference to being a good neighbor to The MathWorks. While I don’t make it a secret of my dislike of many aspects of MATLAB — which I mention later in this article — I do hope they can improve their software and reduce the price. Please note this...

## How to Estimate Encoder Velocity Without Making Stupid Mistakes: Part I

Here's a common problem: you have a quadrature encoder to measure the angular position of a motor, and you want to know both the position and the velocity. How do you do it? Some people do it poorly -- this article is how not to be one of them.

Well, first we need to get position. Quadrature encoders are incremental encoders, meaning they can only measure relative changes in position. They produce a pair of pulse trains, commonly called A and B, that look like...

## How to Read a Power MOSFET Datasheet

One of my pet peeves is when my fellow engineers misinterpret component datasheets. This happened a few times recently in separate instances, all involving power MOSFETs. So it’s time for me to get on my soapbox. Listen up!

I was going to post an article on how to read component datasheets in general. But MOSFETs are a good place to start, and are a little more specific. I’m not the first person to write something about how to read datasheets; here are some other good...

## Thermistor signal conditioning: Dos and Don'ts, Tips and Tricks

In an earlier blog entry, I mentioned this circuit for thermistor signal conditioning:

It is worth a little more explanation on thermistor signal conditioning; it's something that's often done poorly, whereas it's among the easiest applications for signal conditioning.

The basic premise here is that there are two resistors in a voltage divider: Rth is the thermistor, and Rref is a reference resistor. Here Rref is either R3 alone, or R3 || R4, depending on the gain...

## Important Programming Concepts (Even on Embedded Systems) Part I: Idempotence

There are literally hundreds, if not thousands, of subtle concepts that contribute to high quality software design. Many of them are well-known, and can be found in books or the Internet. I’m going to highlight a few of the ones I think are important and often overlooked.

But first let’s start with a short diversion. I’m going to make a bold statement: unless you’re a novice, there’s at least one thing in computer programming about which you’ve picked up...

## Linear Feedback Shift Registers for the Uninitiated, Part VII: LFSR Implementations, Idiomatic C, and Compiler Explorer

The last four articles were on algorithms used to compute with finite fields and shift registers:

- multiplicative inverse
- discrete logarithm
- determining characteristic polynomial from the LFSR output

Today we’re going to come back down to earth and show how to implement LFSR updates on a microcontroller. We’ll also talk a little bit about something called “idiomatic C” and a neat online tool for experimenting with the C compiler.

## Zebras Hate You For No Reason: Why Amdahl's Law is Misleading in a World of Cats (And Maybe in Ours Too)

I’ve been wasting far too much of my free time lately on this stupid addicting game called the Kittens Game. It starts so innocently. You are a kitten in a catnip forest. Gather catnip.

And you click on Gather catnip and off you go. Soon you’re hunting unicorns and building Huts and studying Mathematics and Theology and so on. AND IT’S JUST A TEXT GAME! HTML and Javascript, that’s it, no pictures. It’s an example of an

## Ten Little Algorithms, Part 2: The Single-Pole Low-Pass Filter

Other articles in this series:

- Part 1: Russian Peasant Multiplication
- Part 3: Welford's Method (And Friends)
- Part 4: Topological Sort
- Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method
- Part 6: Green’s Theorem and Swept-Area Detection

I’m writing this article in a room with a bunch of other people talking, and while sometimes I wish they would just SHUT UP, it would be...

## Understanding and Preventing Overflow (I Had Too Much to Add Last Night)

Happy Thanksgiving! Maybe the memory of eating too much turkey is fresh in your mind. If so, this would be a good time to talk about overflow.

In the world of floating-point arithmetic, overflow is possible but not particularly common. You can get it when numbers become too large; IEEE double-precision floating-point numbers support a range of just under 21024, and if you go beyond that you have problems:

for k in [10, 100, 1000, 1020, 1023, 1023.9, 1023.9999, 1024]: try: ...## My Love-Hate Relationship with Stack Overflow: Arthur S., Arthur T., and the Soup Nazi

Warning: In the interest of maintaining a coherent stream of consciousness, I’m lowering the setting on my profanity filter for this post. Just wanted to let you know ahead of time.

I’ve been a user of Stack Overflow since December of 2008. And I say “user” both in the software sense, and in the drug-addict sense. I’m Jason S, user #44330, and I’m a programming addict. (Hi, Jason S.) The Gravatar, in case you were wondering, is a screen...

## Adventures in Signal Processing with Python

Author’s note: This article was originally called Adventures in Signal Processing with Python (MATLAB? We don’t need no stinkin' MATLAB!) — the allusion to The Treasure of the Sierra Madre has been removed, in deference to being a good neighbor to The MathWorks. While I don’t make it a secret of my dislike of many aspects of MATLAB — which I mention later in this article — I do hope they can improve their software and reduce the price. Please note this...

## Chebyshev Approximation and How It Can Help You Save Money, Win Friends, and Influence People

Well... maybe that's a stretch. I don't think I can recommend anything to help you win friends. Not my forte.

But I am going to try to convince you why you should know about Chebyshev approximation, which is a technique for figuring out how you can come as close as possible to computing the result of a mathematical function, with a minimal amount of design effort and CPU power. Let's explore two use cases:

- Amy has a low-power 8-bit microcontroller and needs to compute \( \sqrt{x} \)...

## How to Estimate Encoder Velocity Without Making Stupid Mistakes: Part I

Here's a common problem: you have a quadrature encoder to measure the angular position of a motor, and you want to know both the position and the velocity. How do you do it? Some people do it poorly -- this article is how not to be one of them.

Well, first we need to get position. Quadrature encoders are incremental encoders, meaning they can only measure relative changes in position. They produce a pair of pulse trains, commonly called A and B, that look like...

## Thermistor signal conditioning: Dos and Don'ts, Tips and Tricks

In an earlier blog entry, I mentioned this circuit for thermistor signal conditioning:

It is worth a little more explanation on thermistor signal conditioning; it's something that's often done poorly, whereas it's among the easiest applications for signal conditioning.

The basic premise here is that there are two resistors in a voltage divider: Rth is the thermistor, and Rref is a reference resistor. Here Rref is either R3 alone, or R3 || R4, depending on the gain...

## 10 Software Tools You Should Know

Unless you're designing small analog electronic circuits, it's pretty hard these days to get things done in embedded systems design without the help of computers. I thought I'd share a list of software tools that help me get my job done. Most of these are free or inexpensive. Most of them are also for working with software. If you never have to design, read, or edit any software, then you're one of a few people that won't benefit from reading this.

Disclaimer: the "best" software...

## Ten Little Algorithms, Part 2: The Single-Pole Low-Pass Filter

Other articles in this series:

- Part 1: Russian Peasant Multiplication
- Part 3: Welford's Method (And Friends)
- Part 4: Topological Sort
- Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method
- Part 6: Green’s Theorem and Swept-Area Detection

I’m writing this article in a room with a bunch of other people talking, and while sometimes I wish they would just SHUT UP, it would be...

## Important Programming Concepts (Even on Embedded Systems) Part I: Idempotence

There are literally hundreds, if not thousands, of subtle concepts that contribute to high quality software design. Many of them are well-known, and can be found in books or the Internet. I’m going to highlight a few of the ones I think are important and often overlooked.

But first let’s start with a short diversion. I’m going to make a bold statement: unless you’re a novice, there’s at least one thing in computer programming about which you’ve picked up...

## 10 Circuit Components You Should Know

Chefs have their miscellaneous ingredients, like condensed milk, cream of tartar, and xanthan gum. As engineers, we too have quite our pick of circuits, and a good circuit designer should know what's out there. Not just the bread and butter ingredients like resistors, capacitors, op-amps, and comparators, but the miscellaneous "gadget" components as well.

Here are ten circuit components you may not have heard of, but which are occasionally quite useful.

1. Multifunction gate (

## Byte and Switch (Part 1)

Imagine for a minute you have an electromagnet, and a microcontroller, and you want to use the microcontroller to turn the electromagnet on and off. Sounds pretty typical, right?We ask this question on our interviews of entry-level electrical engineers: what do you put between the microcontroller and the electromagnet?We used to think this kind of question was too easy, but there are a surprising number of subtleties here (and maybe a surprising number of job candidates that were missing...